Gergonne Line

$\Gamma_a$

$\Gamma_b$

$\Gamma_c$

A = (-1.4926844507833976, 2.8151311204675835)
B = (17.51644541767023, 2.2579768178920077)
C = (12.483524467885111, 11.67873073581552)
polígono1, c, a, b = Polygon(A, B, C)
X_1 = TriangleCenter(A, B, C, 1)
polígono2, D, E, F = TrianguloPedal(A, B, C, X_1)
X_3 = TriangleCenter(A, B, C, 3)
d = Line(X_1, X_3)
X_7 = TriangleCenter(A, B, C, 7)
e = Tripolar(A, B, C, X_7)
f = Line(B, C)
g = Line(C, A)
h = Line(A, B)
F' = Intersect(e, h)
D' = Intersect(f, e)
E' = Intersect(g, e)
J = Midpoint(D', D)
k = Circle(J, D)
K = Midpoint(E', E)
p = Circle(K, E)
L = Midpoint(F, F')
q = Circle(L, F)
i = Line(J, K)
X_{3676} = Intersect(e, i)
r: sqrt((b + c - a) * (c + a - b) * (a + b - c) / (a + b + c)) / 2
R: a * b * c / sqrt((a + b + c) * (b + c - a) * (c + a - b) * (a + b - c))
X_{3513} = puntoRazon(X_1, X_3, -2 * r, r + 2 * R + sqrt(r^(2) + 4 * r * R))
X_{3514} = puntoRazon(X_1, X_3, -2 * r, r + 2 * R - sqrt(r^(2) + 4 * r * R))
X_{241} = Intersect(d, e)
X_{3660} = Midpoint(X_{3513}, X_{3514})
s = Circle(A, B, C)
t = Circle(X_1, D)

Tool: TrianguloETC
Dar el vértice A
Input: A_{etc}
Output: B_{etc}, C_{etc}, a, polígono1

c = Circle(A_{etc}, 13)
B_{etc} = Point(c)
d = Circle(A_{etc}, 9)
e = Circle(B_{etc}, 6)
C_{etc} = Intersect(d, e, 1)
polígono1, c_{etc}, a_{etc}, b_{etc} = Polygon(A_{etc}, B_{etc}, C_{etc})
a = Line(B_{etc}, C_{etc})

Tool: Coordenadas Trilineales (command CoordenadasTrilineales)
Dar los vértices de ABC y un  punto P
Input: A, B, C, P
Output: texto2, CoordTrilineales

q_1 = Segment(B, P)
e = Line(A, C)
E_1 = Intersect(e, q_1)
d = Line(C, B)
p_1 = Segment(A, P)
D_1 = Intersect(d, p_1)
ABC, c, a, b = Polygon(A, B, C)
e_1 = OrthogonalLine(P, a)
D = Intersect(e_1, d)
p = Segment(P, D)
α = If(Defined[D_1], -p, p)
f_1 = OrthogonalLine(P, b)
g = OrthogonalLine(P, c)
f = Line(A, B)
E = Intersect(f_1, e)
F = Intersect(g, f)
r_1 = Segment(P, C)
q = Segment(P, E)
r = Segment(P, F)
F_1 = Intersect(f, r_1)
β = If(Defined[E_1], -q, q)
γ = If(Defined[F_1], -r, r)
CoordTrilineales: {α, β, γ}
texto2: "" + CoordTrilineales + ""

Tool: Triangulo Pedal (command TrianguloPedal)
Dar ABC y P
Input: A, B, C, E
Output: polígono2, F, G, H

polígono1, c, a, b = Polygon(A, B, C)
e = OrthogonalLine(E, a)
f = OrthogonalLine(E, b)
g = OrthogonalLine(E, c)
h = Line(B, C)
i = Line(C, A)
j = Line(A, B)
F = Intersect(e, h)
G = Intersect(i, f)
H = Intersect(j, g)
polígono2, h_1, f_1, g_1 = Polygon(F, G, H)

Tool: puntoRazon
Punto X que divide al segmento de extremos P y Q, según una razón  dada, PX : XQ = m : n
Input: P, Q, m, n
Output: X

X = Dilate(Q, m / (m + n), P)

Tool: Circunferencia de Apolonio (command CircunferenciadeApolonio)
Dar los extremos de un segmento y dos números
Input: A, B, m, n
Output: c

N = puntoRazon(A, B, -m, n)
M = puntoRazon(A, B, m, n)
E = Midpoint(N, M)
c = Circle(E, M)

Tool: Tripolar
Dar los vértices de un triángulo y un punto
Input: A, B, C, P
Output: l

d = Line(B, C)
e = Line(C, A)
f = Line(A, B)
g = Line(P, A)
h = Line(P, B)
i = Line(P, C)
E = Intersect(h, e)
F = Intersect(d, g)
j = Line(E, F)
G = Intersect(f, i)
k = Line(G, F)
H = Intersect(e, k)
I = Intersect(j, f)
l = Line(H, I)